Physical significance of nodes?

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SUMMARY

The discussion centers on the physical significance of nodes in quantum mechanics (QM), specifically addressing the concept that nodes, defined as points where particles cannot be located, are not inherently special. Participants emphasize that the energy of the particle determines the presence of nodes, rather than the properties of the spatial points themselves. An analogy to classical physics is provided, comparing the behavior of planets in orbit to the behavior of particles in QM. The conversation also touches on the refinement of Hamiltonians and the non-uniqueness of energy levels in quantum systems.

PREREQUISITES
  • Understanding of quantum mechanics principles
  • Familiarity with Hamiltonian operators in QM
  • Knowledge of eigenstates and eigenvalues
  • Basic concepts of classical physics, particularly Newtonian mechanics
NEXT STEPS
  • Explore the role of Hamiltonian operators in quantum mechanics
  • Study the concept of eigenstates and their significance in QM
  • Investigate the relationship between energy levels and particle behavior in quantum systems
  • Learn about wave functions and their implications for nodes in QM
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Students and professionals in physics, particularly those studying quantum mechanics, as well as educators seeking to clarify the concept of nodes and their implications in quantum systems.

Sheldon Cooper
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Hi,

II have an issue with nodes in QM, like suppose in a well, the number of nodes depend on (n-1), the thing is that, what is so special(physically) about the point that the particle cannot be located there?

Thanks is advance
 
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Sheldon Cooper said:
the thing is that, what is so special(physically) about the point that the particle cannot be located there?
Nothing is special about those points. You can make any point be a node or not be a node just by changing the energy of the particle, which tells us that it's the energy that matters and not some properties of the points in space where the nodes happen to be.

An analogy from classical physics: A planet will pass through some points in space and not others as it moves around the sun. That doesn't make the points along the orbit special; the planet is just going where Newton's laws say it will go and not going anywhere else.
 
Nugatory said:
Nothing is special about those points. You can make any point be a node or not be a node just by changing the energy of the particle, which tells us that it's the energy that matters and not some properties of the points in space where the nodes happen to be.

An analogy from classical physics: A planet will pass through some points in space and not others as it moves around the sun. That doesn't make the points along the orbit special; the planet is just going where Newton's laws say it will go and not going anywhere else.

While I do agree that nodes aren't special, I think you're mistaken about the energy bit, you can always refine a Hamilton by adding cI where I is unity and c is a real constant, this will change all observable energies by c but will not alter the eigenbasis, that is energy is not unique, basis are
 
nashed said:
While I do agree that nodes aren't special, I think you're mistaken about the energy bit, you can always refine a Hamilton by adding cI where I is unity and c is a real constant, this will change all observable energies by c but will not alter the eigenbasis, that is energy is not unique, basis are
Yes, that's right... I could have said ##p^2## to be more accurate.
 
Perfect cancellation of right moving and left moving waves.
 

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